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Let R be an associative ring with identity and ⍀ an n-element set. For k F n consider the R-module M with k-element subsets of ⍀ as basis. The r-stepis the linear map defined on this basis throughwhere the ⌫ are the k y r -element subsets of ⌬.For m -r one obtains chainsof inclusion maps which have interesting homological properties if R has charac-Ž teristic p ) 0. V. B. Mnukhin and J. Siemons J. Combin. Theory 74, 1996 . 287᎐300; J. Algebra 179, 1995, 191᎐199 introduced the notion of p-homology to examine such sequences when r s 1 and here we continue this investigation when r is a power of p. We show that any section of M M not containing certain middle terms is p-exact and we determine the homology modules for such middle terms. Numerous infinite families of irreducible modules for the symmetric groups arise in this fashion. Among these the semi-simple inducti¨e systems discussed by A. Ž . Kleshchev J. Algebra 181, 1996, 584᎐592 appear and in the special case p s 5 we Ž obtain the Fibonacci representations of A. J. E. Ryba J. Algebra 170, 1994, . 678᎐686 . There are also applications to permutation groups of order co-prime to p, resulting in Euler᎐Poincare equations for the number of orbits on subsets of such groups. ᮊ

Ä4 Suppose that ⍀ s 1, 2, . . . , ab for some non-negative integers a and b. Denote Ž . by P a, b the set of unordered partitions of ⍀ into a parts of cardinality b. In this Ž . paper we study the decomposition of the permutation module C P a, b where C is Ž . the field of complex numbers. In particular, we show that C P 3, b is isomorphic to Ž . a submodule of C P b, 3 for b G 3. This settles the next unproven case of a conjecture of Foulkes. ᮊ

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